Aproximações estatísticas para somas de variáveis aleatórias correlacionadas dos tipos Rayleigh e exponencial com aplicação a esquemas de combinação de diversidade

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UNIVERSIDADE ESTADUAL DE CAMPINAS

FACULDADE DE ENGENHARIA EL ÉTRICA E DE COMPUTAÇ ÃO

FRANCISCO RAIMUNDO ALBUQUERQUE PARENTE

STATISTICAL APPROXIMATIONS TO SUMS OF CORRELATED RAYLEIGH AND EXPONENTIAL RANDOM VARIABLES WITH APPLICATION TO

DIVERSITY-COMBINING SCHEMES

APROXIMAÇ ÕES ESTATÍSTICAS PARA SOMAS DE VARI ÁVEIS ALEAT ÓRIAS

CORRELACIONADAS DOS TIPOS RAYLEIGH E EXPONENCIAL COM

APLICAÇ ÃO A ESQUEMAS DE COMBINAÇ ÃO DE DIVERSIDADE

CAMPINAS 2019

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FRANCISCO RAIMUNDO ALBUQUERQUE PARENTE

STATISTICAL APPROXIMATIONS TO SUMS OF CORRELATED RAYLEIGH AND EXPONENTIAL RANDOM VARIABLES WITH APPLICATION TO

DIVERSITY-COMBINING SCHEMES

APROXIMAÇ ÕES ESTATÍSTICAS PARA SOMAS DE VARI ÁVEIS ALEAT ÓRIAS

CORRELACIONADAS DOS TIPOS RAYLEIGH E EXPONENCIAL COM

APLICAÇ ÃO A ESQUEMAS DE COMBINAÇ ÃO DE DIVERSIDADE

Thesis presented to the School of Electrical and Computer Engineering of the University of Campinas in partial fulfillment of the re-quirements for the degree of Master in Elec-trical Engineering, in the area of Telecommu-nications and Telematics.

Disserta¸cão apresentada à Faculdade de En-genharia Elétrica e de Computa¸cão da Uni-versidade Estadual de Campinas como parte dos requisitos exigidos para obten¸cão do t´ıtulo de Mestre em Engenharia Elétrica, na ´

area de Telecomunica¸c˜oes e Telem´atica.

ADVISOR/ORIENTADOR: PROF. DR. JOS ´E C ˆANDIDO SILVEIRA SANTOS FILHO

ESTE TRABALHO CORRESPONDE A`

VERS ÃO FINAL DA DISSERTAÇ ÃO

DE-FENDIDA PELO ALUNO FRANCISCO

RAIMUNDO ALBUQUERQUE PARENTE

E ORIENTADA PELO PROF. DR. JOS ´E

C ˆANDIDO SILVEIRA SANTOS FILHO.

CAMPINAS 2019

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Agência(s) de fomento e nº(s) de processo(s): CAPES ORCID: https://orcid.org/0000-0001-9827-6707

Ficha catalográfica

Universidade Estadual de Campinas Biblioteca da Área de Engenharia e Arquitetura Elizangela Aparecida dos Santos Souza - CRB 8/8098

Parente, Francisco Raimundo Albuquerque,

P215s ParStatistical approximations to sums of correlated Rayleigh and exponential random variables with application to diversity-combining schemes / Francisco Raimundo Albuquerque Parente. – Campinas, SP : [s.n.], 2019.

ParOrientador: José Cândido Silveira Santos Filho.

ParDissertação (mestrado) – Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação.

Par1. Canal em desvanecimento. 2. Correlação (Estatística). 3. Distribuição assintótica (Teoria da probabilidade). 4. Sistemas de comunicação sem fio. 5. Variáveis aleatórias. I. Santos Filho, José Cândido Silveira, 1979-. II.

Universidade Estadual de Campinas. Faculdade de Engenharia Elétrica e de Computação. III. Título.

Informações para Biblioteca Digital

Título em outro idioma: Aproximações estatísticas para somas de variáveis aleatórias

correlacionadas dos tipos Rayleigh e exponencial com aplicação a esquemas de combinação de diversidade

Palavras-chave em inglês:

Fading channel Correlation (Statistics)

Asymptotic distribution (Probability theory) Wireless communication systems

Random variables

Área de concentração: Telecomunicações e Telemática Titulação: Mestre em Engenharia Elétrica

Banca examinadora:

Michel Daoud Yacoub Gustavo Fraidenraich

Edgar Eduardo Benítez Olivo

Data de defesa: 30-01-2019

Programa de Pós-Graduação: Engenharia Elétrica

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COMISS ÃO JULGADORA - DISSERTAÇ ÃO DE MESTRADO

Candidato: Francisco Raimundo Albuquerque Parente RA: 190748 Orientador: Prof. Dr. Jos´e Cˆandido Silveira Santos Filho

Data de Defesa: 30 de janeiro de 2019

T´ıtulo da Disserta¸cão: “Statistical Approximations to Sums of Correlated Rayleigh and Exponential Random Variables with Application to Diversity-Combining Schemes” (Aproxima¸cões Estat´ısticas para Somas de Variáveis Aleatórias Correlacionadas dos Tipos Rayleigh e Exponencial com Aplica¸cão a Esquemas de Combina¸cão de Diversidade)

Membros da Comiss˜ao Julgadora:

Prof. Dr. Michel Daoud Yacoub (Presidente, UNICAMP) Prof. Dr. Gustavo Fraidenraich (Titular Interno, UNICAMP) Prof. Dr. Edgar Eduardo Ben´ıtez Olivo (Titular Externo, UNESP)

A ata de defesa, com as respectivas assinaturas dos membros da Comissão Julgadora, encontra-se no SIGA (Sistema de Fluxo de Disserta¸cão/Tese) e na Secretaria de Pós-Gradua¸cão da Faculdade de Engenharia Elétrica e de Computa¸cão.

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AGRADECIMENTOS

Este trabalho n˜ao seria poss´ıvel sem o suporte e o carinho de pessoas brilhantes com as quais tive a oportunidade de conviver e para as quais dedico esta Disserta¸c˜ao.

Ao meu orientador, Prof. Dr. José Cândido Silveira Santos Filho, cujos brilhantismo genu´ıno, dedica¸cão e companheirismo têm me inspirado como pesquisador e como pessoa. O aprendizado e a motiva¸cão que tive nesses anos de trabalho conjunto embasaram a qualidade deste projeto e de muitos outros que hão de vir.

Aos meus professores da pós-gradua¸cão e membros da banca examinadora, Prof. Dr. Michel Daoud Yacoub e Prof. Dr. Gustavo Fraidenraich, pela constante solicitude e pelas inestimáveis contribui¸cões a este projeto e a muitos outros ao longo do Mestrado.

Ao Prof. Dr. Edgar Eduardo Ben´ıtez Olivo, pela participa¸cão na banca examinadora e pelos comentários e sugestões que engrandecem este trabalho.

Ao Prof. Dr. Daniel Benevides da Costa, pela constante aten¸cão e pelo cordial aux´ılio durante minha mudan¸ca para a UNICAMP ao término da gradu¸cão na UFC.

`

A CAPES, pelo apoio financeiro, e `a UNICAMP, que me propiciou vislumbrar um horizonte superior com as virtudes ´eticas e profissionais nela presentes.

Aos meus amigos, principalmente aos que conheci desde a minha chegada ao WissTek Laboratory, pelos valiosos conselhos e momentos de descontra¸c˜ao nas horas de folga. Com vocˆes, a jornada se tornou muito mais prazerosa e gratificante.

Aos meus pais, Raimundo e Antonia, e à minha irmã, Thalya, por todo apoio, carinho e amor dedicados. Vocês são meu alicerce, minha fortaleza e meu bem maior. Amo-os imensamente.

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ABSTRACT

Sums of random variables are widely applied to wireless communications systems. Exam-ples include linear equalization, signal detection, interference phenomena, and diversity-combining schemes. However, the exact formulation for the statistical functions of these sums, such as the probability density function and the cumulative distribution function, requires in general a complicated mathematical treatment, which has motivated the search for simple approximate solutions. Although there are several approximate proposals avail-able in the literature, many of which obtained through the traditional moment-matching technique, they do not offer a good fit under the regime of high signal-to-noise ratio. It is well-known that this regime is a paramount region for the performance analysis of communications systems in terms of important metrics such as bit error rate and outage probability. More recently, in order to circumvent this limitation, a new promising tech-nique known as asymptotic matching was proposed, capable of providing approximations for statistics of the sum of random variables with an excellent fit under the regime of high signal-to-noise ratio. Even so, this technique was initially proposed for the sum of mutually independent variables only, and thus it has not been applicable to sums of cor-related variables. In this work, a novel asymptotic analysis is proposed, from which it is possible to generalize the application of asymptotic matching to the correlated case. The proposed analysis is illustrated for sums of Rayleigh and sums of exponential variables with arbitrary correlation and arbitrary fading parameters. Furthermore, closed-form asymptotic expressions are derived in order to obtain new simple and precise approxima-tions under the regime of high signal-to-noise ratio. As application examples, practical diversity-combining schemes are addressed, namely, equal-gain combining and maximal-ratio combining. Finally, numerical results show the excellent performance of the proposed approximations in comparison to the approximations obtained via moment matching.

Keywords: Asymptotic analysis, correlation, diversity combining, fading channels, sums of random variables.

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RESUMO

Somas de variáveis aleatórias são amplamente aplicadas em sistemas de comunica¸cão sem fio. Exemplos incluem equaliza¸cão linear, deteçcão de sinais, fenômenos de inter-ferência e esquemas de combina¸cão de diversidade. No entanto, a formula¸cão exata para as fun¸cões estat´ısticas dessas somas, como a fun¸cão densidade de probabilidade e a fun¸cão distribui¸cão acumulada, requer em geral um tratamento matemático complicado, o que tem motivado a busca por solu¸cões aproximadas mais simples. Apesar de haver várias propostas de aproxima¸cão dispon´ıveis na literatura, muitas das quais obtidas usando-se a tradicional técnica de casamento de momentos, elas não oferecem um bom ajuste em regime de alta rela¸cão sinal-ru´ıdo. Sabe-se, porém, que essa é uma região primordial para a análise de desempenho de sistemas de comunica¸cão em termos de métricas importantes como taxa de erro de bit e probabilidade de interrup¸cão. Mais recentemente, com o intuito de contornar essa limita¸cão, foi proposta uma nova técnica promissora conhecida como casamento de ass´ıntotas, capaz de fornecer aproxima¸cões para estat´ısticas de somas de variáveis aleatórias positivas com um ótimo ajuste em regime de alta rela¸cão sinal-ru´ıdo. Ainda assim, essa técnica foi inicialmente implementada apenas para o caso de somas de variáveis independentes, não sendo até então aplicável para somas de variáveis cor-relacionadas. Neste trabalho, uma nova análise assintótica é proposta, a partir da qual é poss´ıvel generalizar o uso do casamento de ass´ıntotas para o caso correlacionado. A análise proposta é ilustrada para somas de variáveis Rayleigh e somas de variáveis exponenciais com correla¸cão e parâmetros de desvanecimento arbitrários. Além disso, deduzem-se ex-pressões assintóticas em forma fechada com o intuito de obter novas aproxima¸cões simples e precisas em regime de alta rela¸cão sinal-ru´ıdo. Como exemplos de aplica¸cão, esquemas práticos de combina¸cão de diversidade são abordados, quais sejam, combina¸cão por ganho igual e combina¸cão por razão máxima. Por fim, resultados numéricos mostram o excelente desempenho das aproxima¸cões propostas em compara¸cão com as aproxima¸cões obtidas via casamento de momentos.

Palavras-chave: Análise assintótica, canais de desvanecimento, combina¸cão de diversi-dade, correla¸cão, somas de variáveis aleatórias.

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LIST OF FIGURES

Figure 1 - Sum statistics of two correlated Rayleigh RVs. . . 38

Figure 2 - Sum statistics of three correlated Rayleigh RVs. . . 39

Figure 3 - Sum statistics of four correlated Rayleigh RVs. . . 40

Figure 4 - Sum statistics of two correlated exponential RVs. . . 42

Figure 5 - Sum statistics of three correlated exponential RVs. . . 43

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LIST OF ABBREVIATIONS AND ACRONYMS

BER Bit-Error Rate

CF Characteristic Function

cdf Cumulative Distribution Function EGC Equal-Gain Combining

5G Fifth Generation

MRC Maximal-Ratio Combining OP Outage Probability

pdf Probability Density Function RV Random Variable

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LIST OF SYMBOLS

˜

X random variable that approximates a random variable X ∼ asymptotically equal to (around zero)

Ω average power

Ωi average power of the ith component (or branch)

J0(·) Bessel function of the first kind and zeroth order

ΦX(·) characteristic function of a random variable X

∗ convolution operator ρ correlation coefficient

FX(·) cumulative distribution function of a random variable X

dB decibel

, definition operator det(·) determinant operator

di,j distance between ith and jth antennas

≡ equivalence operator exp(·) exponential function F {·} Fourier transform

gi gain at the ith MRC diversity branch

Γ(·) gamma function

j imaginary unit number (equals√−1) F−1_{·} _{inverse Fourier transform}

Wi ith exponential random variable

Ri ith Rayleigh random variable

E[Xk] kth moment of a random variable X K−1 inverse of a matrix K

N mean noise power

fX(·) multivariate probability density function of a vector random variable X

M amount of random variables in the sum REGC output of the equal-gain combiner

RM RC output of the maximal-ratio combiner

˜

µ parameter of the gamma and α-µ approximate distributions ˜

Ω parameter of the gamma, Nakagami-m, Weibull, and α-µ approximate distributions

σi parameter of the ith Rayleigh RV

˜

m parameter of the Nakagami-m approximate distribution m parameter of the Nakagami-m distribution

˜

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α parameter of the α-µ distribution µ parameter of the α-µ distribution

fX(·) probability density function of a random variable X

N set of natural numbers

Γi signal-to-noise ratio at the ith MRC diversity branch

Γ signal-to-noise ratio at the MRC output N total noise power

(·)T _{transpose operator}

V[·] variance operator λ carrier wavelength

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LIST OF PUBLICATIONS

• F. R. A. Parente e J. C. S. Santos Filho, “Aproxima¸cões estat´ısticas para so-mas de variáveis Rayleigh correlacionadas e aplica¸cão,” Anais do XXXVI Simpósio Brasileiro de Telecomunica¸cões e Processamento de Sinais (SBrT’18), Campina Grande, Brasil, Set. 2018, pp. 563–567.

• F. R. A. Parente and J. C. S. Santos Filho, “Asymptotically exact framework to ap-proximate sums of positive correlated random variables and application to diversity-combining receivers,” IEEE Wireless Communications Letters, in press.

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CONTENTS

1 INTRODUCTION . . . 15

1.1 Literature Review and Motivation . . . 16

1.2 Contributions . . . 17

1.3 Structure . . . 18

2 SUMS OF RANDOM VARIABLES . . . 19

2.1 Problem Formulation . . . 19 2.2 Exact Solutions . . . 19 2.2.1 Independent Case . . . 20 2.2.2 Correlated Case . . . 21 2.3 Approximation Techniques . . . 21 2.3.1 Moment Matching . . . 22 2.3.2 Asymptotic Matching . . . 22 3 PROPOSED APPROXIMATIONS . . . 24 3.1 Preliminaries . . . 24

3.2 Sums of Correlated Rayleigh Random Variables . . . 25

3.2.1 Exact Sum Statistics . . . 26

3.2.2 Weibull Approximation . . . 28

3.2.3 Nakagami-m Approximation . . . 29

3.2.4 α-µ Approximation . . . 29

3.2.5 Application to Equal-Gain Combining (EGC). . . 30

3.3 Sums of Correlated Exponential Random variables . . . 31

3.3.1 Exact Sum Statistics . . . 31

3.3.2 Weibull Approximation . . . 32

3.3.3 Gamma Approximation. . . 33

3.3.4 α-µ Approximation . . . 33

3.3.5 Application to Maximal-Ratio Combining (MRC) . . . 34

4 NUMERICAL RESULTS . . . 36

4.1 Sums of Rayleigh Random Variables. . . 37

4.2 Sums of Exponential Random Variables . . . 41

5 CONCLUSIONS . . . 45

5.1 Final Considerations . . . 45

5.2 Future Work. . . 46

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15

Chapter

1

INTRODUCTION

Nowadays people live surrounded by electronic devices that keep them continuously interconnected. It is well-known that wireless technologies have disrupted mobile commu-nications, bringing altogether the world to a new technological stage. In fact, due recent developments and research, mobile devices are increasingly more robust and efficient, whose performance is comparable to that of fixed stations.

Nevertheless, the wireless environment is chaotic by nature. The channel itself may drastically distorts the propagation signal, which undergoes path loss and several other phenomena, such as scattering, reflection, and diffraction [1, 2]. Due to such phenomena, the signal reaches the receiver with a large number of scattered, reflected, and diffracted waves, coming from diverse paths, with random amplitudes and phases, generating what is called multipath propagation. The combination of these factors stochastically alters both amplitude and phase of the received signal, an effect known as fading [2, 3].

One way to overcome the limitations imposed by fading in wireless systems consists of using diversity-combining schemes [3, 4]. Basically, these schemes provide the receiver with multiple replicas (branches) of the transmitted signal, which are then combined to obtain a resulting signal of better quality.

There are several types of diversity-combining techniques, such as equal-gain com-bining (EGC) and maximal-ratio comcom-bining (MRC). These two schemes are additive, and therefore their performance analysis in terms of bit-error rate (BER) and outage probability (OP) requires knowledge of sum statistics, namely, the probability density function (pdf) or, equivalently, the cumulative distribution function (cdf). However, the exact computation of these statistics is rather cumbersome, since it involves a multifold integration over the multivariate pdf of the summands [5]. As the number of random variables (RVs) in the sum increases, the exact formulation may prove unfeasible, which has motivated the search for simple approximate solutions.

Sums of RVs can be applied not only to diversity combining but also to many other communications schemes, such as signal detection and linear equalization. Due to its

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Chapter 1. INTRODUCTION 16

importance, several works have proposed approximations to sums considering a variety of fading scenarios. On this concern, the next section presents a brief review of literature with the main research achievements to date and the motivation for this work.

1.1 Literature Review and Motivation

Since the first approximations for sums of RVs were proposed by Nakagami [6], re-searchers have tried to find good approximate solutions for a variety of fading scenarios. For instance, some works proposed approximations to the sum of non-identical inde-pendent Nakagami-m RVs by using either the Nakagami-m distribution itself [7] or the generalized α-µ distribution [8]. Some accurate approximations have also been obtained for sums of many other distributions, such as the sum of Ricean [9] and α-µ [10] RVs.

Nonetheless, many approximations proposed in the literature have been obtained under the constraint that the summands are mutually independent and by using the traditional moment-matching technique [7–10]. Such technique has been designed to provide a good fit in the distribution body, but it loses track of the distribution tail. This region corresponds to the regime of high signal-to-noise ratio (SNR), which is a compelling scenario to compare different communications systems in terms of BER and OP.

In order to overcome this drawback of moment-based approximations, it has been recently proposed a new approach known as asymptotic matching [11]. In this technique, the asymptote of the approximate distribution is matched to the asymptote of the exact sum distribution, guaranteeing an outstanding fit at the high-SNR regime. Even though asymptotic matching offers better approximations at the distribution tail, its use is very recent and has been limited to the independent case only.

More recently, several works have addressed correlated fading scenarios, which are a more realistic assumption to model emerging communication techniques over massive multiple-input multiple-output systems [12, 13]. In such systems, some undesirable corre-lation between the input-output links may arise due to insufficiently spaced antennas [14]. For instance, considering some diversity-combining schemes over particular fading distri-butions, asymptotic expressions to approximate performance metrics in the high-SNR regime were derived in [15–21]. Specifically, it was observed in [16–21] that the asymp-totic system performance over the correlated channels addressed therein is a scaled version of the asymptotic system performance over independent channels, with the scale factor depending on the correlation matrix. Interestingly, though, it has been overlooked so far that a broad class of positive correlated RVs behaves asymptotically as an equivalent set of mutually independent RVs, which is an insightful and fundamental result explored herein. Another very recent work [22] aimed to approximate the body of the distribution of sums of correlated Weibull RVs by using expressions in terms of the Meijer G-function.

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However, as highlighted in [23], this approach may notably depart from the exact dis-tribution tail, and even lead to computationally erroneous results near the origin. This region corresponds to the important regime of high SNR, as one can move toward the distribution tail either by reducing the value of the instantaneous SNR or by increasing the value of the average SNR [16].

From the above reasons, it is important to obtain new accurate approximations for the challenging correlated scenario, specially in the high-SNR region, of most practical interest. In this way, capitalizing on a new asymptotic result for sums of correlated RVs, we propose a unified, general approach to design approximations that render an excellent fit at the high-SNR regime (i.e., at the cdf tail). Particularly, we investigate sums of correlated Rayleigh RVs and sums of correlated exponential RVs with arbitrary fading parameters in both cases. Various candidate approximate distributions are proposed and discussed. As application examples, we analyze the performance of EGC and MRC operating over correlated Rayleigh fading channels. These and other contributions of this work are outlined next.

1.2 Contributions

In this work, the following contributions are provided:

(i) Capitalizing on a new fundamental result elaborated herein, the asymptotic-matching scheme is extended to the correlated scenario, allowing for accurate statistical ap-proximations to general sums near the origin, or, equivalently, at high SNR.

(ii) Asymptotically optimal approximations are proposed to sums of Rayleigh RVs and sums of exponential RVs with arbitrary correlation and arbitrary fading parameters. These approximations keep a good track of the body of the exact sum distribution while ensuring an outstanding fit at the distribution tail, i.e., at high SNR.

(iii) For comparison purposes, the performance of some candidate approximate distri-butions are evaluated, namely, Nakagami-m, gamma, Weibull, and α-µ distridistri-butions.

(iv) New simple, closed-form, asymptotic expressions are derived and applied to EGC and MRC schemes operating over correlated Rayleigh fading.

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1.3 Structure

The remainder of this work is organized as follows.

Chapter 2: This chapter introduces the problem formulation for sums of RVs.

Considering both independent and correlated fading scenarios, we revisit the ex-act solution to the problem as well as some approximate approaches available in the literature.

Chapter 3: A newly fundamental insight on sums of arbitrarily correlated RVs is

introduced in this chapter. Capitalizing on this novel result, asymptotic matching is performed in order to provide optimal approximations around the origin to sums of correlated Rayleigh and exponential RVs. Various candidate approximate distri-butions are presented. Finally, the analysis is applied to analyze output statistics of two different diversity-combining schemes, namely, EGC and MRC.

Chapter 4: This chapter illustrates the excellent performance of the proposed

approximations for many scenarios. The exact sum statistics are approximated by the Nakagami-m, gamma, Weibull, and α-µ distributions. Numerical results show that the new approximations outclass conventional moment-based approximations, especially at high SNR.

Chapter 5: The main conclusions are summarized in this chapter. It is also

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19

Chapter

2

SUMS OF RANDOM VARIABLES

There are several applications of sums of RVs in wireless communications, such as signal detection, linear equalization, and diversity-combining schemes. In these scenarios, the evaluation of system performance in terms of BER and OP requires knowledge of the sum pdf or the sum cdf, whose exact formulation may prove unfeasible. This chapter introduces the exact general solution to find the statistics of sums of RVs, addressing both independent and correlated cases. Afterwards, two methods that provide approximate solutions to circumvent the intricacy of the exact approach are discussed, namely, the traditional moment-matching and the new asymptotic-matching techniques.

2.1 Problem Formulation

Let S be the sum of M arbitrarily correlated RVs Si, i ∈ {1, . . . , M }, i.e.,

S =

M

X

i=1

Si. (1)

The problem consists of finding the sum pdf and the sum cdf of S, denoted as fS(·)

and FS(·), respectively. As the cdf can be determined from the pdf in a straightforward

manner, the analysis herein is developed based on the pdf alone.

2.2 Exact Solutions

The general formulation to obtain the exact sum pdf fS(·) of S requires knowledge

of the multivariate pdf fS(·) , fS1,...,SM(·, . . . , ·) of S , [S1· · · SM] T

. In this section, the analysis to obtain the exact sum statistics of arbitrarily correlated RVs is described. Previously, though, the independent scenario is revisited.

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Chapter 2. SUMS OF RANDOM VARIABLES 20

2.2.1 Independent Case

For the particular case of mutually independent RVs, the exact sum pdf fS(·) of S is

given by either the convolution of the marginal pdfs fSi(·) or the inverse Fourier transform

of the product of the individual characteristic functions (CFs) ΦSi(·) of Si [24].

The first approach is the multidimensional convolution of the marginal pdfs fSi(·)

of the summands, i.e.,

fS(s) = fS1(s1) ∗ fS2(s2) ∗ · · · ∗ fSM(sM). (2)

Let the characteristic function ΦSi(·) of Si be defined as [24]

ΦSi(ω) ,

Z ∞

−∞

fSi(si) exp(jωsi)dsi, (3)

where j , √−1 is the imaginary unit. Note from (3) that the characteristic function ΦSi(·) of Si can be viewed as the Fourier transform of the pdf fSi(·) of Si (with a reversal

in the sign of the exponent), i.e.,

ΦSi(ω) =F {fSi(si)}. (4)

In this way, taking the Fourier transform of (2), it yields

ΦS(ω) = M

Y

i=1

ΦSi(ω). (5)

Furthermore, from the Fourier transform inversion formula, the pdf fSi(·) of Si is given

by [24] fSi(si) = 1 2π Z ∞ −∞ ΦSi(ω) exp(−jωsi)dω, (6)

and it follows that

fSi(si) =F −1

{ΦSi(ω)}. (7)

Note from (4) and (7) that the pdf fSi(·) and the CF ΦSi(·) of Si form a unique Fourier

transform pair. This approach provides another way to obtain the statistics of the sum S. For instance, assuming knowledge of the CF ΦSi(·) of each RV Si, the exact sum pdf fS(·)

of S can be attained by taking the inverse Fourier transform of (5), i.e.,

fS(s) =F−1{ΦS(ω)}. (8)

Therefore, when the RVs are mutually independent, (2) and (8) provide two ways to obtain the exact pdf fS(·) of the sum S. However, when the RVs are mutually correlated,

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these approaches cannot be applied. In this case, a formulation known as Brennan’s integral should be used instead, which is described next.

2.2.2 Correlated Case

Considering the scenario when the summands are positive RVs, it was shown in [5] by using a geometric approach that the pdf fS(·) and the cdf FS(·) of the sum S can be

formulated as fS(s) = Z s 0 Z s−sM 0 · · · Z s−PM_i=3si 0 fS s − M X i=2 si, s2, . . . , sM ! ds2· · · dsM −1dsM (9a) FS(s) = Z s 0 Z s−sM 0 · · · Z s−PM_i=3si 0 Z s−PM_i=2si 0 fS(s1, s2, . . . , sM) ds1ds2· · · dsM −1dsM. (9b)

The integral in (9) is known as Brennan’s integral, which is a general formulation to obtain the exact pdf fS(·) and cdf FS(·) of the sum of either independent or correlated

RVs. Note that fS(·) and FS(·) are expressed as a multidimensional integral of the

multi-variate pdf fS(·). Therefore, even though Brennan’s formulation is general and exact, it

provides closed-form solutions only for particular cases. Furthermore, its implementation in computing softwares may prove unfeasible when the number M of summands increases (e.g., M > 5).

In order to circumvent this limitation, many approximate approaches have been proposed in the literature. In Section 2.3, two methods to provide approximate solutions for both independent and correlated scenarios are presented.

2.3 Approximation Techniques

In this section, two approaches used to approximate the exact sum distribution of either independent or correlated summands are covered. Initially, the classical moment-matching technique is presented. Then, a more recent approach known as asymptotic matching is discussed. In both cases, we assume that a certain candidate distribution f_S˜(·) has been selected to approximate the exact sum. So the only remaining task is to

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2.3.1 Moment Matching

A well-known approach used to approximate the statistics of sums of RVs is called moment matching [7–10]. In this method, some moments of the exact sum S are matched to the corresponding moments of the candidate approximate RV ˜S, i.e.,

E[ ˜Sk] = E[Sk], (10)

where E[ ˜Sk_{] is the kth moment of the approximate distribution, and E[S}k_{] is the kth}

moment of the exact sum distribution, k ∈ N. Particularly, should the RVs in the sum be independent, E[Sk_{] can be obtained from the individual moments of the summands}

as [7, eq. (6)] E[Sk] = k X k1=0 k1 X k2=0 · · · kM −2 X kM −1=0 k k1 k1 k2 · · ·kM −2 kM −1 E[S1k−k1]E[S k1−k2 2 ] · · · E[S kM −1 M ]. (11)

However, should the RVs in the sum be mutually correlated and the CF of S be known, E[Sk] can be obtained from the moment theorem as [24]

E[Sk] = 1 jk dk dωkΦS(ω) ω=0 . (12)

Moment-based approximations guarantee a good fit mainly in the distribution body. On the other hand, it loses track of the distribution tail at high-SNR regime, which is a compelling scenario to compare different communication systems. To overcome such limitation, a new method called asymptotic matching has been proposed, which is pre-sented next.

2.3.2 Asymptotic Matching

Assuming a scenario where the summands are mutually independent, an approach known as asymptotic matching has been recently proposed [11]. In this method, the parameters of the approximate distribution are adjusted so that its asymptote equals the asymptote of the exact sum distribution.

Let the Maclaurin series expansion of the marginal pdf fSi(·) of Si be given by

fSi(si) = ∞

X

n=0

ai,nsibi,n ∼ ai,0s bi,0

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and the Maclaurin series expansion of the sum pdf fS(·) of S be expressed by

fS(s) = ∞

X

n=0

ansbn ∼ a0sb0, (14)

where the symbol “∼” denotes “asymptotically equal to (around zero)”. Since the sum pdf is expressed as the multidimensional convolution of the marginal pdfs in the independent case, the asymptote (around the origin) a0sb0 of fS(·) in (14) is the multidimensional

convolution of the M corresponding asymptotes ai,0sibi,0 of each marginal pdf in (13).

More specifically, it is shown in [11] (using a similar procedure as in the proof sketch in [16, Proposition 4]) that a0 and b0 are given by

a0 = M Y i=1 ai,0Γ (bi,0+ 1) Γ M + M X i=1 bi,0 ! (15a) b0 = (M − 1) + M X i=1 bi,0, (15b)

where Γ(·) denotes the gamma function. Note that the sum’s asymptotic parameters (a0 and b0) are given exclusively in terms of the number of summands (M ) and their

marginal asymptotic parameters (ai,0 and bi,0, i ∈ {1, . . . , M }). Moreover, in a log-scale

plot, note from (14) that a0 and b0 determine the linear and angular coefficients of the

asymptote of the sum pdf, respectively.

In order to perform asymptotic matching, the parameters of the approximate pdf are adjusted so that its asymptote, say f_S˜(·) ∼ ˜a0s

˜

b0_{, equals the asymptote of the exact}

sum pdf in (14). This is achieved by forcing

˜

a0 = a0 (16a)

˜b0 = b0. (16b)

Assuming the distribution parameters of each summand are known, we can then ad-just the parameters of the approximate distribution by solving the system of equations in (16). This matching guarantees that both the exact and approximate distributions are asymptotically the same, providing an excellent fit around the origin, i.e., at high SNR.

Since the asymptotic-matching approach has been proposed under the independent constraint, its use is in principle not applicable to the correlated case. However, due to a novel insight on sums of correlated RVs introduced in Chapter 3, this technique can be exploited in the correlated scenario as well.

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24

Chapter

3

PROPOSED APPROXIMATIONS

Several works in the literature have attempted to accurately approximate sums of RVs. Considering the case of arbitrarily correlated summands, this task has proven even more challenging. Some recent works (cf. [15–21]) derived asymptotic expressions to ap-proximate performance metrics of diversity-combining schemes operating over correlated fading scenarios. Although capable of describing the system performance for particular fading scenarios, these results are neither general nor distribution-oriented. In this chapter a new general framework is introduced in order to design approximate distributions that well fit the whole body of the exact sum distribution while being asymptotically exact near the origin. The analysis is then applied to the sum of correlated Rayleigh and expo-nential RVs. For illustrative purposes, new simple, closed-form, asymptotic expressions are derived and applied to the performance analysis of two classical diversity-combining techniques, namely, EGC and MRC.

3.1 Preliminaries

Let us assume that the asymptote of the multivariate pdf of S can be expressed in the form fS(s) ∼ a M Y i=1 sbi i , (17)

where a and bi are constants.1 This implies that the asymptote of the sum pdf can be

expressed as a multidimensional convolution, i.e.,

fS(s) ∼ a sb1 1 ∗ s b2 2 ∗ · · · ∗ s bM M . (18)

1_{This is a mild condition that holds true for many popular fading models, such as the Rayleigh, Rice,}

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Chapter 3. PROPOSED APPROXIMATIONS 25

We can restate (17) in a more convenient form, i.e.,

fS(s) ∼ M Y i=1 ˆ ai,0s ˆ bi,0 i , (19) where ˆ ai,0 , a 1 M (20a) ˆbi,0 , bi. (20b)

Note the implications raised by (19). We can view the ith term ˆai,0s ˆ bi,0

i in the

product as the asymptote of an equivalent marginal pdf. And the product of these M terms is asymptotically equal to the multivariate pdf of S. In other words, (19) implies that, around the origin, the correlated RVs Si in the sum behaves as an equivalent set of

mutually independent RVs. Accordingly, the asymptote a0sb0 of the sum pdf in (14) is

given by the convolution of the M equivalent marginal asymptotes ˆai,0s ˆ_b_i,0

i in (19). This

is a novel and general asymptotic result for sums of positive correlated RVs, with many further implications. For instance, we can apply (15) (obtained for the independent case) to the correlated scenario. To this end, we just replace ai,0 and bi,0 in (15) by ˆai,0 and ˆbi,0

in (20), respectively, so as to determine a0 and b0 for the correlated case.

Once the asymptote a0sb0 of the sum pdf is determined, we can match it with the

asymptote ˜a0s ˜

b0 _{of the approximate pdf, i.e., we can force (16). This guarantees an}

asymp-totically optimal fit in the high-SNR regime. Furthermore, as a candidate approximate distribution may have more than two parameters to be adjusted, and as the asymptotic matching provides only two equations, it may be necessary to use asymptotic match-ing along with other existmatch-ing methods. Since the moment matchmatch-ing can provide a good approximation in the body of the pdf, we propose its use in order to complement the asymptotic matching. In this way, when the approximate distribution has l > 2 parame-ters, (10) can provide the remaining l − 2 equations to complete the system of equations and find the distribution parameters accordingly.

Our proposed analysis can be used for designing statistical approximations to sums of a broad class of positive correlated RVs. As a case study, next we investigate two kinds of correlated sums, namely, sums of Rayleigh RVs and sums of exponential RVs.

3.2 Sums of Correlated Rayleigh Random Variables

In this section, we initially discuss the sums of correlated Rayleigh RVs in order to apply our analysis. Thereafter, some candidate distributions are provided to approximate

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the exact sum. Even though our framework is suitable for a variety of candidate distri-butions, we illustrate the development by using the generalized, versatile α-µ distribution and two of its particular cases, namely, Nakagami-m (α = 2, µ = m) and Weibull (µ = 1) distributions [25]. The analysis can then be used to evaluate the performance of an EGC scheme operating over correlated Rayleigh fading, as discussed at the end of the session.

3.2.1 Exact Sum Statistics

Let R (≡ S) be the sum of M arbitrarily correlated Rayleigh RVs Ri (≡ Si),

i ∈ {1, . . . , M }, i.e., R = M X i=1 Ri. (21)

The marginal pdf of each RV Ri is given by

fRi(ri) = ri σ2 i exp − r 2 i 2σ2 i , ri ≥ 0, (22)

where σi > 0 is a scale parameter, and Ωi , E[R2i] = 2σi2 is the average power. In order

to specify the multivariate Rayleigh pdf fR(·) , fR1,...,RM(·, . . . , ·) of R , [R1· · · RM] T

, it is appropriate to decompose each RV in terms of its in-phase and quadrature compo-nents, i.e.,

Ri =

q X2

i + Yi2, (23)

where Xiand Yi are independent and identically distributed Gaussian RVs for each i, with

zero mean and variance V[Xi] = V[Yi] = σi2 [26]. Note that in general (Xi,Xj), (Yi,Yj),

and (Xi,Yj) are pairs of correlated RVs, i 6= j. We can arrange the components Xi and

Yi into the vector form

X , [X1· · · XM]T and Y , [Y1· · · YM]T , (24)

so that their marginal and joint statistics can be specified by the covariance matrix of X, the covariance matrix of Y , and the cross-covariance matrix between X and Y — KXX , E[XXT], KY Y , E[Y YT], and KXY , E[XYT], respectively. These

three matrices can then be rearranged into a unique (symmetric and non-singular) matrix defined as K , " KXX KXY KT XY KY Y # . (25)

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Hence, the multivariate Rayleigh pdf can be expressed as a function of the matrix K only [26] fR(r) = M Y i=1 ri (2π)M_[det(K)]12 Z π −π · · · Z π −π exp −1 2g (r, φ) dφ1· · · dφM, (26)

where r , [r1· · · rM]T ∈ [0, ∞)M, φ , [φ1· · · φM]T ∈ [−π, π)M, and g (r, φ) is given

by [26]

g (r, φ) =

M

X

i=1

(Aiicos2φi+ Ciisin2φi+ 2Biicos φisin φi)r2i

+

M

X

i,j=1 i6=j

(Aijcos φicos φj + Cijsin φisin φj+ 2Bijcos φisin φj)rirj, (27)

with Aij, Bij, and Cij being obtained from

A , KXX− KXYK_{Y Y}−1 KXYT −1 (28) B , − KXX− KXYKY Y−1 K T XY −1 KXYKY Y−1 (29) C , KY Y − KXYT K −1 XXKXY −1 . (30)

Using the Maclaurin series expansion of the integrand in (26) and then taking its first term, the asymptote of the multivariate Rayleigh pdf is obtained as

fR(r) ∼ M Y i=1 ri [det(K)]12 . (31)

Therefore, from (31), we have for the Rayleigh case that

a = 1 [det(K)]12

(32a)

bi = 1. (32b)

Replacing ai,0 and bi,0 by ˆai,0 and ˆbi,0 in (15), respectively, and using the results from (20)

and (32), we obtain a0 = 1 [det(K)]12Γ(2M ) (33a) b0 = 2M − 1. (33b)

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In order to apply moment matching, we can use the first moment E [R] of the sum, which is sufficient for the scope of this work and is easily obtained from (21) and (22) as

E[R] =pπ/2

M

X

i=1

σi. (34)

Finally, using (33) and (34), we can obtain approximations to the exact sum distri-bution by performing the matching techniques accordingly. Once performed, the matching techniques provide the parameters of the approximate distribution in terms of those of the exact sum distribution. Hence, one can adjust the approximate pdf/cdf by properly setting its parameters. This is illustrated next for three different approximations.

3.2.2 Weibull Approximation

In the first proposed approximation, the sum R of correlated Rayleigh RVs is ap-proximated by a Weibull RV ˜R, whose pdf is given by [27, eq. (4-43)]

f_R˜(r) = ˜ αrα−1˜ ˜ Ω exp −r ˜ α ˜ Ω , (35)

where ˜α > 0 is the shape (fading) parameter, and ˜Ω = E[ ˜Rα˜_{] is the scale parameter of the}

distribution. Our objective is to find the values of the parameters ˜α and ˜Ω of the Weibull pdf such that f_R˜(·) renders a good approximation to the exact sum pdf fR(·).

In order to guarantee a good adjustment at the high-SNR regime, one can per-form asymptotic matching. To this end, taking the Maclaurin series expansion of the exponential function in (35), the coefficients ˜a0 and ˜b0 can be obtained as

˜ a0 = ˜ α ˜ Ω (36a) ˜b0 = ˜α − 1. (36b)

Finally, substituting (33) and (36) into (16), and solving the system of equations for the parameters ˜α and ˜Ω, we obtain

˜

α = 2M (37a)

˜

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3.2.3 Nakagami-m Approximation

In the second proposed approximation, the sum R of correlated Rayleigh RVs is approximated by a Nakagami-m RV ˜R, whose pdf is expressed by [6, eq. (3)]

f_R˜(r) = 2 ˜mm˜_r2 ˜m−1 Γ( ˜m) ˜Ωm˜ exp −mr˜ 2 ˜ Ω , (38)

where ˜Ω = E[ ˜R2_{] and ˜}_{m , ˜}_Ω2_{/V[ ˜}_R2_{] are the parameters of the distribution. Similarly}

as for the previous Weibull approximation, our objective is to find the values of the parameters ˜m and ˜Ω of the Nakagami-m pdf such that f_R˜(·) renders a good approximation

to the exact sum pdf fR(·).

Hence, taking the Maclaurin series expansion of the exponential function in (38), the coefficients ˜a0 and ˜b0 can be obtained as

˜ a0 =

2 ˜mm˜

Γ( ˜m) ˜Ωm˜ (39a)

˜b0 = 2 ˜m − 1. (39b)

Substituting (33) and (39) into (16), and solving the system of equations for the param-eters ˜m and ˜Ω, we obtain

˜ m = M (40a) ˜ Ω = M ( 2[det(K)]12Γ(2M ) Γ(M ) )_M1 . (40b) 3.2.4 α-µ Approximation

Since both Weibull and Nakagami-m pdfs have only two parameters, the asymptotic matching itself is sufficient to solve the system of equations. Nevertheless, in order to obtain more degrees of freedom, it is important to investigate distributions containing more than two parameters. For illustrative purposes, we depict this case by using the generalized α-µ distribution, whose pdf is [25]

f_R˜(r) = ˜ α˜µµ˜_rα˜˜µ−1 Γ(˜µ) ˜Ωµ˜ exp −µr˜ ˜ α ˜ Ω , (41)

where ˜α > 0, ˜Ω = E[ ˜Rα˜_{], and ˜}_{µ , ˜}_Ω2_{/V[ ˜}_Rα˜_{] are the parameters of the distribution.}

The analysis here is similar to that of the two previous approximations, except that now we have one more parameter. In this way, one more equation is needed, which can

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be provided by the moment-matching technique. Therefore, from (10) and (16), it is necessary to obtain ˜a0, ˜b0, and E[ ˜Rk] of the approximate distribution in order to perform

the matching techniques accordingly.

Initially, by taking the Maclaurin series expansion of the exponential function in (41), the coefficients ˜a0 and ˜b0 are easily obtained as

˜ a0 = ˜ α˜µµ˜ Γ(˜µ) ˜Ωµ˜ (42a) ˜b0 = ˜α˜µ − 1, (42b)

and its kth moment is given by [25]

E[ ˜Rk] = ˜ Ωkα˜Γ k ˜ α + ˜µ ˜ µαk˜Γ(˜µ) . (43)

As the first moment is sufficient for the scope of this work, we can set k = 1 in (43), which gives E[ ˜R] = ˜ Ωα1˜Γ 1 ˜ α + ˜µ ˜ µα1˜Γ(˜µ) . (44)

These results can then be combined into a set of three transcendental equations by substituting (33), (34), (42), and (44) into (10) and (16). Even though there is no closed-form solution for this set of equations, one can in principle solve it numerically by using a computing software such as Mathematica or MATLAB, obtaining the parameters of the approximate pdf in terms of those of the exact sum pdf.

3.2.5 Application to Equal-Gain Combining (EGC)

The proposed analysis can be directly applied to the study of diversity-combining schemes. As an application example, we investigate the EGC technique operating over correlated Rayleigh fading channels.

Considering an EGC scheme with M arbitrarily correlated Rayleigh fading branches Ri, one can express its output envelope REGC by [1]

REGC = 1 √ M M X i=1 Ri = R √ M, (45)

where √M is a normalization factor that accounts for the increased output noise. The EGC output in (45) is simply the sum in (21) normalized by √M . For simplicity, we drop the normalization, as this is just a scale factor that can be handled through a trivial

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transformation of variables. Therefore, the analysis based on the sum R in (21) is directly applicable to the EGC output REGC in (45).

By applying the proposed analysis to EGC schemes operating over correlated Rayleigh fading, one can obtain approximate pdfs and cdfs to the EGC output. In the high-SNR regime, these approximations are asymptotically optimal and can then be used to evaluate the EGC performance in terms of BER and OP.

3.3 Sums of Correlated Exponential Random variables

Similarly as for the Rayleigh case above, this section introduces sums of correlated exponential RVs in order to apply our analysis. For illustrative purposes, the exact sum distribution is approximated by the α-µ distribution and two of its particular cases, namely, gamma (α = 1) and Weibull distributions [25]. As detailed at the end of the section, the analysis can be readily applied to evaluate the performance of an MRC scheme operating over correlated Rayleigh fading.

3.3.1 Exact Sum Statistics

Let W (≡ S) be the sum of M arbitrarily correlated exponential RVs Wi (≡ Si),

i ∈ {1, . . . , M }, i.e., W = M X i=1 Wi, (46)

where the exponential RV is defined as Wi , R2i. The multivariate exponential pdf

fW(·) , fW1,...,WM(·, . . . , ·) of W , [W1· · · WM] T _{is expressed by [26]} fW(γ) = 1 (4π)M_[det(K)]12 Z π −π · · · Z π −π exp −1 2h (γ, φ) dφ1· · · dφM, (47)

where γ , [γ1· · · γM]T ∈ [0, ∞)M, and h (γ, φ) is given by [26]

h (γ, φ) =

M

X

i=1

(Aiicos2φi+ Ciisin2φi+ 2Biicos φisin φi)γi

+

M

X

i,j=1 i6=j

(Aijcos φicos φj + Cijsin φisin φj+ 2Bijcos φisin φj)(γiγj) 1

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with Aij, Bij, and Cij defined, as before, from (28)–(30).

Taking the Maclaurin series expansion of the integrand in (47), it is straightforward to show that the asymptote of the multivariate exponential pdf is expressed by

fW(γ) ∼

1

2M_[det(K)]1₂. (49)

Hence, from (49), we have for the exponential case that

a = 1

2M_[det(K)]1₂ (50a)

bi = 0. (50b)

Replacing ai,0 and bi,0 by ˆai,0 and ˆbi,0 in (15), respectively, and using the results from (20)

and (50), we obtain a0 = 1 2M_[det(K)]12Γ(M ) (51a) b0 = M − 1. (51b)

Considering the average Rayleigh power defined in Section 3.2, i.e., Ωi , E[R2i] =

E[Wi], the first moment E[W ] of the sum is easily obtained as

E[W ] =

M

X

i=1

Ωi. (52)

Using (51) and (52), one can attain approximations to the exact sum distribution by performing the matching techniques in a similar manner as in Section 3.2. In fact, the development detailed therein is general and readily applicable to sums of exponential RVs as well, as illustrated next.

3.3.2 Weibull Approximation

In order to approximate the sum W of correlated exponential RVs by a Weibull RV ˜

W , one can follow the procedure described in Subsection 3.2.2. In this way, taking the Maclaurin series expansion of the Weibull pdf in (35), the coefficients ˜a0 and ˜b0 required

for asymptotic matching are given by (36).

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for the parameters ˜α and ˜Ω, we obtain

˜

α = M (53a)

˜

Ω = 2M_{M [det(K)]}1₂_{Γ(M ).} _(53b)

3.3.3 Gamma Approximation

As the exponential RV is a squared Rayleigh RV, it is interesting to investigate the approximation of the sum W (of correlated exponential RVs) not by the Nakagami-m RV as in Subsection 3.2.3, but by its squared version. The squared Nakagami-m RV is a gamma RV ˜W [25], whose pdf f_W˜(·) is [27, eq. (4-34)]

f_W˜(γ) = ˜ µµ˜_γµ−1˜ Γ(˜µ) ˜Ωµ˜ exp −µγ˜ ˜ Ω , (54)

where ˜Ω = E[ ˜W ] and ˜µ , ˜Ω2_{/V[ ˜}_{W ] are the parameters of the distribution.}

Using the Maclaurin series expansion of the exponential function in (54), the coef-ficients ˜a0 and ˜b0 can be obtained as

˜ a0 = ˜ µµ˜ Γ(˜µ) ˜Ωµ˜ (55a) ˜b0 = ˜µ − 1. (55b)

Finally, substituting (51) and (55) into (16), and solving the system of equations for the parameters ˜Ω and ˜µ, we obtain

˜

Ω = 2M [det(K)]2M1 (56a)

˜

µ = M. (56b)

3.3.4 α-µ Approximation

In order to provide more degrees of freedom during the adjustment process, one can choose an approximate distribution with more than two parameters. For illustrative purposes, we depict this case in the same way as in Subsection 3.2.4, where the generalized α-µ distribution was used. The analysis here is similar to the previous one.

The α-µ pdf has three parameters, namely, ˜α, ˜µ, and ˜Ω. Hence, three equations are necessary to solve the system of equations and find the distribution parameters properly. As for the asymptotic-matching step, the coefficients ˜a0 and ˜b0 given by (42) are used.

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first moment of the α-µ distribution, which is given by (44).

Therefore, one can combine these results into a set of three transcendental equa-tions by substituting (42), (44), (51), and (52) into (10) and (16). Although there is no closed-form solution for this system of equations, it can be solved numerically by using a computing software, obtaining the parameters of the approximate pdf in terms of those of the exact sum pdf.

3.3.5 Application to Maximal-Ratio Combining (MRC)

The analysis developed on sums of correlated exponential RVs is of great importance in wireless communications. For instance, it can be applied to study the performance anal-ysis of the MRC technique operating over correlated Rayleigh fading channels, as follows. Let us assume an MRC scheme consisting of M arbitrarily correlated Rayleigh fading branches Ri. Its output RM RC can be expressed by [1]

RM RC = M

X

i=1

giRi, (57)

where gi is the gain at the ith branch. An important performance measure is the SNR

Γi, defined for each branch i as

Γi ,

local mean signal power

mean noise power , (58)

where the local mean signal power is given by R2

i/2 [1]. Assuming the presence of Gaussian

noise with mean power Ni = N in each branch, then

Γi =

R2 i

2N. (59)

The total noise power N at the MRC output is given by N = N M X i=1 g2 i. (60)

Hence, the resulting SNR Γ is

Γ = R 2 M RC 2N = 1 2 PM i=1giRi 2 NPM i=1gi2 . (61)

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Furthermore, it is shown in [1] that the SNR Γ is maximized if each gain gi is equal to

the ratio of the signal voltage to noise power of the respective branch, i.e.,

gi =

Ri

N. (62)

Therefore, substituting (62) into (61), it follows that

Γ = 1 2 PM i=1R2i/N 2 NPM i=1(Ri/N )2 = M X i=1 R2 i 2N = M X i=1 Γi. (63)

The result in (63) shows that the output SNR Γ of the MRC is the sum of the SNRs Γi

in each branch.

Comparing (63) to (46), note that Γi ≡ Wi and Γ ≡ W . Consequently, the analysis

developed based on the sum W in (46) is also applicable to the MRC output SNR Γ in (63). To this end, by applying the proposed analysis to MRC schemes operating over correlated Rayleigh fading, one can obtain approximate pdfs and cdfs to the MRC output SNR. In the high-SNR regime, these approximations are asymptotically optimal and can then be used to evaluate the MRC performance in terms of BER and OP.

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36

Chapter

4

NUMERICAL RESULTS

This chapter presents several numerical results in order to evaluate the performance of the statistical approximations proposed in this work. We compare our approximations with those obtained by using the traditional moment-based approach (cf. [7–10]). The exact solution shown in the plots has been computed by numerically integrating Brennan’s formula, reproduced in (9).

We present curves of pdf and cdf for sums of correlated Rayleigh and sums of correlated exponential RVs. The pdfs are shown in terms of envelope level for Rayleigh sums and in terms of power level for exponential sums. The cdfs are plotted in terms of average power (per branch) for both sums, since this is a common practice in the literature and shows the high-SNR regime. As for the combining techniques discussed in Chapter 3, the distribution of the sum envelope level corresponds to the distribution of the EGC output, and the distribution of the sum power level corresponds to the distribution of the MRC output. For illustrative purposes, we let

E[XiYj] = 0, ∀i, j, (64a)

E[XiXj] = E[YiYj] = ρi,j, ∀i 6= j. (64b)

Note that ρi,j is the correlation coefficient between the ith and jth components [24], which

can be modeled by ρi,j = J0 2πdi,j λ , (65)

where J0(·) is the Bessel function of the first kind and zeroth order, di,j is the distance

between ith and jth antennas, and λ is the wavelength of the carrier signal. For simplicity, we fix ρi,j = ρ, ∀i, j, where ρ ∈ {0.1, 0.5, 0.9}. All curves are in a log-scale plot and were

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Chapter 4. NUMERICAL RESULTS 37

4.1 Sums of Rayleigh Random Variables

Initially, we consider the sum of correlated Rayleigh RVs. Figures 1, 2, and 3 show the exact and approximate pdfs and cdfs of the sum of two, three, and four RVs, respec-tively. For the pdf curves, we fix the average power per input branch at Ωi = 2σi2 = 2,

∀i, and vary the sum envelope level r. For the cdf curves, we fix the sum envelope level at r = 1 and vary the average power per input branch Ωi = Ω, ∀i.

We provide three candidate distributions to approximate the exact sum, namely, the Nakagami-m, Weibull, and α-µ distributions. These approximations are obtained by using two different approaches: our proposed analysis and moment matching alone. On the one hand, as for our proposed approach, only asymptotic matching is required for obtaining the Nakagami-m and Weibull approximations, whereas the proposed α-µ approximation uses asymptotic matching along with the first moment of the sum (to perform the matching step). On the other hand, in order to obtain the moment-based approximations, we have chosen for simplicity (i) the first and second moments for the Nakagami-m and Weibull approximations, and (ii) the first, second, and third moments for the α-µ approximation.

From the pdf plots, note that our approximations are asymptotically optimal, match-ing the exact curve near the origin — at pdf left tail. In this region, our proposed Nakagami-m, Weibull, and α-µ approximations outclass the moment-based Nakagami-m, Weibull, and α-µ counterparts, respectively. Particularly, the proposed Nakagami-m and Weibull approximations near the origin outperform the proposed α-µ approximation as M and ρ increase. This shows that, in the region of most practical interest, the two approx-imations that use asymptotic matching alone provide better results than the proposed α-µ approximation, where moment matching was introduced. In fact, there is a trade-off between asymptotic-matching and moment-matching: the former provides a better fit in the left pdf tail — as expected by design —, and the latter offers a better fit in the right pdf tail. However, the left pdf tail is of most practical interest when comparing different communications systems in terms of BER and OP, since it corresponds to the quintessential high-SNR regime.

Similar results can be noticed from the cdf plots. In this case, the cdf right tail cor-responds to the high-SNR regime, where our proposed approximations are asymptotically optimal. When applied to the EGC scheme, these cdf curves reveal that our proposed approximations keep track of the diversity (slope) and coding (offset) gains of the exact combining output, clearly outperforming the moment-based approximations.

(38)

Chapter 4. NUMERICAL RESULTS 38 ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △○○○ ○○○○○○○○○_○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○○○○ ○ ○ ○ ○ ○ ○ ○ □□□□□□□□□□_□ □ □ □ □□□□□□□□□□□□□ □ □ □ □ □ □ △△△△△△△△△△△△△ △ △ △ △ △ △ △ △△△△△△△△△△△△△ △ △ △ △ △ △ △ △ Exact

○ Proposed Nakagami-m approximation ○ Moment-based Nakagami-m approximation □ Proposed Weibull approximation □ Moment-based Weibull approximation △ Proposed -μ approximation △ Moment-based -μ approximation -40 -30 -20 -10 0 10 20 30 104 102 100 10-2 10-4 10-6 10-8 Envelope, r dB) pdf , fR r) ○ ○ ○ ○ _○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ _□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ _□ □ □ □ □ □ _□ □ □ □ □ □ △ △ △ △ _△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ _△ △ △ △ △ △ △ △ △ △ △ △ Exact

○ Proposed Nakagami-m approximation ○ Moment-based Nakagami-m approximation □ Proposed Weibull approximation □ Moment-based Weibull approximation △ Proposed -μ approximation △ Moment-based -μ approximation -10 0 10 20 30 100 10-2 10-4 10-6 10-8 10-10 10-12 10-14

Average power per branch, Ω dB

cdf

,F

R

r=

1

(a) Pdf of the sum with ρ = 0.1. (b) Cdf of the sum with ρ = 0.1.

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ ○○○○○○○○○○○○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○○○_○ ○ ○ ○ ○ ○ ○ ○ ○ □□□□□□□□□□ □ □ □ □ □□□□□□□□□□□□□_□ □ □ □ □ □ □ △△△△△△△△△△△△△_△ △ △ △ △ △ △ △ △△△△△△△△△△△△_△△ △ △ △ △ △ △ △ △ △ Exact

○ Proposed Nakagami-m approximation ○ Moment-based Nakagami-m approximation □ Proposed Weibull approximation □ Moment-based Weibull approximation △ Proposed -μ approximation △ Moment-based -μ approximation -40 -30 -20 -10 0 10 20 30 104 102 100 10-2 10-4 10-6 10-8 Envelope, r dB) pdf , fR r) ○ ○ ○ ○ _○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ _○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ _□ □ _□ □ _□ □ _□ □ □ _□ □ □ △ △ △ △ _△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ _△ △ △ △ △ △ △ △ △ △ △ △ Exact

cdf

,F

R

r=

1

(c) Pdf of the sum with ρ = 0.5. (d) Cdf of the sum with ρ = 0.5.

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △○○○○○○○○○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○○○○○_○ ○ ○ ○ ○ ○ ○ ○ □□□□□□ □ □ □ □ □ □□□□□□□□□□□□□□ □_□ □ □ □ □ □ □ △△△△△△△△△△△△△_△△ △ △ △ △ △ △ △ △ △ △ △△△△△△△△△△△△△ △△_△ △ △ △ △ △ △ △ Exact

○ Proposed Nakagami-m approximation ○ Moment-based Nakagami-m approximation □ Proposed Weibull approximation □ Moment-based Weibull approximation △ Proposed -μ approximation △ Moment-based -μ approximation -40 -30 -20 -10 0 10 20 30 104 102 100 10-2 10-4 10-6 10-8 Envelope, r dB) pdf , fR r) ○ ○ ○ ○ ○ _○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ _○ ○ _○ ○ _○ ○ ○ _○ ○ ○ _○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ _□ □ _□ □ _□ □ _□ □ _□ □ _□ △ △ △ △ △ _△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ _△ △ _△ △ _△ △ △ _△ △ _△ △ Exact

cdf

,F

R

r=

1

(e) Pdf of the sum with ρ = 0.9. (f) Cdf of the sum with ρ = 0.9.

(39)

Chapter 4. NUMERICAL RESULTS 39 ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △○○ ○○○○ ○○○○○○○○○_○ ○ ○ ○ ○ ○ ○ ○○○○ ○○○○○○○○○○○○_○ ○ ○ ○ ○ ○ □□□□ □□□□ □□□_□ □ □ □ □□□□□□ □□□□□□□□□□ □ □ □ □ △△△△ △△△△△△△△△△△△△ △ △ △ △ △ △△△△ △△△△△△△△△△△△△ △ △ △ △ △ △ Exact

○ Proposed Nakagami-m approximation ○ Moment-based Nakagami-m approximation □ Proposed Weibull approximation □ Moment-based Weibull approximation △ Proposed -μ approximation △ Moment-based -μ approximation -40 -30 -20 -10 0 10 20 30 104 102 100 10-2 10-4 10-6 10-8 Envelope, r dB) pdf , fR r) ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ _□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ Exact

cdf

,F

R

r=

1

(a) Pdf of the sum with ρ = 0.1. (b) Cdf of the sum with ρ = 0.1.

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △○○ ○○○○ ○○○○○○○○ ○ ○ ○ ○ ○ ○ ○ ○○○○ ○○○○○○○○○○○○_○ ○ ○ ○ ○ ○ ○ ○ □□□□ □□□□□□□ □ □ □ □□□□□□□□□□□□□□□□□ □ □ □ □ □ △△△△ △△△△△△△△△△△△_△ △ △ △ △ △ △ △ △△△△ △△△△△△△△△△△△△ △ △ △ △ △ △ △ △ △ Exact

○ Proposed Nakagami-m approximation ○ Moment-based Nakagami-m approximation □ Proposed Weibull approximation □ Moment-based Weibull approximation △ Proposed -μ approximation △ Moment-based -μ approximation -40 -30 -20 -10 0 10 20 30 104 102 100 10-2 10-4 10-6 10-8 Envelope, r dB) pdf , fR r) ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ _□ □ □ □ □ _□ □ □ □ □ □ □ □ △ △ △ _△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ _△ △ △ △ △ △ △ △ △ △ △ △ △ Exact

cdf

,F

R

r=

1

(c) Pdf of the sum with ρ = 0.5. (d) Cdf of the sum with ρ = 0.5.

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △○○ ○○○○○○○○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○○○○○○○○○○_○○ ○ ○ ○ ○ ○ ○ ○ ○ □□□□□□□ □ □ □□□□□□□□□□□□□□□□□_□ □ □ □ □ □ □ □ △△△△△△△△△△△△△△△△△△ △_△ △ △ △ △ △ △ △ △ △△△△△△△△△△△△△△△△△_△ △ △ △ △ △ Exact

○ Proposed Nakagami-m approximation ○ Moment-based Nakagami-m approximation □ Proposed Weibull approximation □ Moment-based Weibull approximation △ Proposed -μ approximation △ Moment-based -μ approximation -40 -30 -20 -10 0 10 20 30 104 102 100 10-2 10-4 10-6 10-8 Envelope, r dB) pdf , fR r) ○ ○ ○ ○ _○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ _○ ○ ○ _○ ○ ○ _○ ○ ○ ○ ○ _○ ○ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ _□ □ _□ □ _□ □ _□ □ _□ □ _□ □ _□ △ △ △ _△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ _△ △ △ △ △ _△ △ △ △ △ △ _△ △ Exact

cdf

,F

R

r=

1

(e) Pdf of the sum with ρ = 0.9. (f) Cdf of the sum with ρ = 0.9.